Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Write the binomial expansion for each expression. $$(x+y)^{6}$$

Step-by-Step Solution

Answer

The expansion is \( x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 \).

Step 1: Identify the General Formula

Recall the binomial theorem, which states that: \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). In this case, \( a = x \), \( b = y \), and \( n = 6 \).

Step 2: Calculate Binomial Coefficients

Determine the binomial coefficients using \( \binom{6}{k} \) for \( k = 0, 1, 2, 3, 4, 5, 6 \). These values are \( \binom{6}{0} = 1 \), \( \binom{6}{1} = 6 \), \( \binom{6}{2} = 15 \), \( \binom{6}{3} = 20 \), \( \binom{6}{4} = 15 \), \( \binom{6}{5} = 6 \), and \( \binom{6}{6} = 1 \).

Step 3: Plug Coefficients into the Formula

Substitute each binomial coefficient and the powers of \( x \) and \( y \) into the expansion formula. This will give the expanded terms: \( \binom{6}{0} x^{6}y^{0} + \binom{6}{1} x^{5}y^{1} + \binom{6}{2} x^{4}y^{2} + \binom{6}{3} x^{3}y^{3} + \binom{6}{4} x^{2}y^{4} + \binom{6}{5} x^{1}y^{5} + \binom{6}{6} x^{0}y^{6} \).

Step 4: Compute Each Term

Compute each term using the binomial coefficients found: \( 1x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + 1y^6 \).

Step 5: Write the Final Expansion

Combine all calculated terms to form the final expanded result: \( x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6 \).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful tool in algebra that allows us to expand expressions raised to a power. At its core, the binomial expansion lets us express the powers of a binomial, such as \((x+y)^n\), as a sum of terms of the form \(a^nb^k\). Each term in the expansion is comprised of three key components:

• A coefficient from the binomial coefficients
• The variable \(a\) raised to a power decreasing by \(k\)
• The variable \(b\) raised to the power \(k\)

For example, when expanding \((x+y)^6\), the expression uses the coefficients provided by the binomial theorem to create terms like \(x^6\) and \(6xy^5\). Each element of the expansion requires careful computation, arranged in increasing powers of the second term \(b\). The sum of the powers of \(x\) and \(y\) in each term will always add up to the original power \(n\).
The binomial expansion is particularly helpful in simplifying calculations for large powers, making complex equations more manageable.

Binomial Coefficients

In a binomial expansion, binomial coefficients play a crucial role. These coefficients, derived from Pascal's Triangle, determine the weight of each term in the expansion. For example, in the expansion of \((x+y)^6\), the sequence of coefficients is extracted using the formula \(\binom{n}{k}\), where \(n\) is the power of the binomial and \(k\) is the term position:

• \(\binom{6}{0} = 1\)
• \(\binom{6}{1} = 6\)
• \(\binom{6}{2} = 15\)
• \(\binom{6}{3} = 20\)
• \(\binom{6}{4} = 15\)
• \(\binom{6}{5} = 6\)
• \(\binom{6}{6} = 1\)

These coefficients can be calculated directly using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where "!" denotes the factorial of a number. They indicate the number of ways to choose \(k\) items from \(n\) items, which in the context of binomial expansion, maps out the distribution of terms. Understanding binomial coefficients is essential for writing the full expansion and dictates how each composed term interacts in the polynomial expression.

Polynomial Expansion

Polynomial expansion refers to expressing a product of terms expanded into a summed form. For binomials such as \((x+y)^6\), polynomial expansion transforms this compact form into a string of summed terms, each a result of multiplying terms iteratively using the binomial theorem. Start by calculating each term using binomial coefficients, followed by the exponential management of the terms:

• First term: Multiply the highest power of the first term \(x\) with the lowest power of the second term \(y\), i.e., \(x^6\)
• Ensure throughout that the sum of the exponents in every term equals the original binomial power \(n\), which is 6 in this case
• Continue until the powers of \(y\) are maximized in the last term, i.e., \(y^6\)

The goal of polynomial expansion is to uncover each possible term derived from the initial binomial raised to a power. Skills in polynomial expansion are foundational for increasing ease in handling more complex mathematical formulations and applications.